Code

library(needs)
needs(igraph,
      igraphdata,
      netseg)
data("UKfaculty")

Relational Sociology and the concept of embeddedness

Hier Theorie Beschreibung und die verschiedenen Ansätze

Text on all the different approaches to embeddedness and social capital etc.

Reciprocity

Measures of reciprocity are based on the dyad census: Any pair of vertices in a network is called a dyad. There are three possible states of dyads (in a directed network): null dyads (no edge), b) asymmetric dyads (one directed edge) and c) mutual dyads (two directed edges).

Code

null <- graph_from_literal(1, 2)
as <- graph_from_literal(1-+2)
mutual <- graph_from_literal(1-+2, 2-+1)

g <- graph_from_literal(1-+2, 2-+3, 2+-3, 5-+2, 3--+5, 5-+3, 3-+4)

layout_line <- cbind(1:2, rep(1, 2))

plot(null,
     vertex.color = "lavender",
     vertex.size = 30, 
     layout = layout_line)

plot(as, 
     vertex.color = "lavender",
     vertex.size = 30,
     layout = layout_line)

plot(mutual,
     vertex.color = "lavender",
     vertex.size = 30,
     layout = layout_line)

# Plotte das Netzwerk mit Lavender vertices
plot(g,
     vertex.color = "lavender",
     vertex.size = 30,
     edge.width = 2,
     arrow.size = 0.1,
     layout = layout_with_fr)

null	asymmetric	mutual
5	3	2

Dyad census in a graph

Code

UKfaculty <- upgrade_graph(UKfaculty)
UKfaculty <- simplify(UKfaculty)

dyad_census(UKfaculty)

The level of reciprocity shows whether a network mainly consists of mutual or asymetrical edges and can be calculated in R using either ‘default’-mode (number of reciprocated edges divided by the total number of edges) or ‘ratio’-mode (number of mutual dyads divided by the number of asymmetric dyads).

Code

reciprocity(UKfaculty, mode="default")

reciprocity(UKfaculty, mode="ratio")

Transitivity

Any set of 3 vertices is called a triad. In directed networks, 16 different triads exist (Davis and Leinhardt 1967).

Code

# Define all 16 triad types using adjacency matrices
triad_matrices <- list(
  matrix(c(0,0,0, 0,0,0, 0,0,0), nrow=3, byrow=TRUE), # 003
  matrix(c(0,1,0, 0,0,0, 0,0,0), nrow=3, byrow=TRUE), # 012
  matrix(c(0,1,0, 1,0,0, 0,0,0), nrow=3, byrow=TRUE), # 102
  matrix(c(0,1,1, 0,0,0, 0,0,0), nrow=3, byrow=TRUE), # 021D
  matrix(c(0,0,0, 1,0,0, 1,0,0), nrow=3, byrow=TRUE), # 021U
  matrix(c(0,1,0, 0,0,1, 0,0,0), nrow=3, byrow=TRUE), # 021C
  matrix(c(0,1,0, 1,0,0, 1,0,0), nrow=3, byrow=TRUE), # 111D
  matrix(c(0,1,0, 1,0,1, 0,0,0), nrow=3, byrow=TRUE), # 111U
  matrix(c(0,1,1, 0,0,0, 0,1,0), nrow=3, byrow=TRUE), # 030T
  matrix(c(0,1,0, 0,0,1, 1,0,0), nrow=3, byrow=TRUE), # 030C
  matrix(c(0,1,1, 1,0,0, 1,0,0), nrow=3, byrow=TRUE), # 201
  matrix(c(0,1,1, 0,0,1, 0,1,0), nrow=3, byrow=TRUE), # 120D
  matrix(c(0,0,0, 1,0,1, 1,1,0), nrow=3, byrow=TRUE), # 120U
  matrix(c(0,1,1, 0,0,1, 1,0,0), nrow=3, byrow=TRUE), # 120C
  matrix(c(0,0,1, 1,0,1, 1,1,0), nrow=3, byrow=TRUE), # 210
  matrix(c(0,1,1, 1,0,1, 1,1,0), nrow=3, byrow=TRUE) # 300 (mutual all)
)

triad_names <- c("003", "012", "102", "021D", "021U", "021C", "111D", "111U", "030T", "030C",
                 "201", "120D", "120U", "120C", "210", "300")

# Plot each triad
par(mfrow = c(4, 4), mar = c(1, 1, 2, 1))  # 4x4 grid layout

for (i in 1:16) {
  adj <- triad_matrices[[i]]
  g <- graph_from_adjacency_matrix(adj, mode = "directed")
  plot(
    g,
    main = triad_names[i],
    vertex.color = "lavender",
    vertex.size = 30,
    vertex.label = NA,
    vertex.label.cex = 0,
    edge.arrow.size = 0.3,
    layout = layout_in_circle
  )
}

Triads 201 are called intransitive
Triads 300 are called transitive
Triads 003 and 012 are called vacuously transitive

The codes like 102, 021C, 120U may feel cryptic at first, but they follow a logical pattern:

** Basic structure: Three-digit code

First digit: Number of mutual (reciprocal) dyads (e.g., A ↔︎ B)
Second digit: Number of asymmetric (one-way) dyads (e.g., A → B)
Third digit: Number of null dyads (no tie between the pair)

Example: ‘102’

Digit	Meaning
`1`	1 mutual (reciprocal) relationship (e.g., A ↔︎ B)
`0`	0 one-way relationships
`2`	2 node pairs have no relationship

Add-On lettering:

For more complex cases, a letter is added to describe the triad’s shape or direction:

Code	Description
`021D`	Two one-way edges going down (A → B → C)
`021U`	Two one-way edges going up (C → B → A)
`021C`	A circle of one-way edges (A → B → C → A)
`030T`	A transitive triangle (A → B → C and A → C)
`030C`	A cycle with three one-way edges (A → B → C → A)
`120D`	One mutual edge, two one-way edges in a downward direction
`120U`	One mutual edge, two one-way edges in an upward direction
`120C`	A circle-like pattern with mutual + one-way ties
`210`	Two mutual dyads and one one-way edge
`300`	All three dyads are mutual (fully connected triad)

We can compute the triad census of a network using

Code

triad_census(UKfaculty)
# nice layout
tbl <- data.frame(t(triad_census(UKfaculty)))
colnames(tbl) <- c("003", "012", "102", "021D", "021U", "021C", "111D", "111U", "030T", "030C", "201", "120D", "120U", "120C", "210", "300")
tbl

In undirected networks only 4 distinct triads exist.

Code

# Define the 4 undirected triad types
undirected_triad_matrices <- list(
  matrix(c(0,0,0, 0,0,0, 0,0,0), nrow=3, byrow=TRUE),  # 0 edges: Empty (Type: 003)
  matrix(c(0,1,0, 1,0,0, 0,0,0), nrow=3, byrow=TRUE),  # 1 edge: One dyad (Type: 102)
  matrix(c(0,1,1, 1,0,0, 1,0,0), nrow=3, byrow=TRUE), # 2 edges: Open triad (Type: 201)
  matrix(c(0,1,1, 1,0,1, 1,1,0), nrow=3, byrow=TRUE)  # 3 edges: Complete triad (Type: 300)
)

undirected_triad_names <- c("003", "102", "201", "300")

# Plot each undirected triad
par(mfrow = c(2, 2), mar = c(1, 1, 2, 1))  # 2x2 grid layout

for (i in 1:4) {
  adj <- undirected_triad_matrices[[i]]
  graph <- graph_from_adjacency_matrix(adj, mode = "undirected")
  plot(
    graph,
    main = undirected_triad_names[i],
    vertex.color = "lavender",
    vertex.label = NA,
    vertex.size = 30,
    vertex.label.cex = 0,
    layout = layout_in_circle
    )
}

Network segregation and homophily

Segregation: network level property-Segregation refers to the extent to which actors and social groups are separated from another. Segregation exists if the share of ties between groups is much less pronounced than the share of in-group ties, or if the groups are fully separated from each other.

Segregation in the Schelling model (preference = 55%)

The literature does not agree regarding the exact terminology of homophily and segregation. To some authors, homophily refers to both the macro-phenomenon (lack of intergroup relationships) and the micro-mechanism (similarity preferences). To others, homophily refers to the macro-phenomenon and homophilic or homophilious selection refer to the micro-mechanism.

I concur with other researchers on the terminology of homophily (preferences) for the mechanism and segregation for the macro-pattern.

Code

# Check the attributes available in the dataset
vertex_attr_names(UKfaculty)

# Assuming 'Group' is an attribute in the dataset
# Assign different colors to each unique group
group_colors <- rainbow(length(unique(V(UKfaculty)$Group)))

# Create a vector of colors for each vertex based on its group
vertex_colors <- group_colors[V(UKfaculty)$Group]

# Plot the graph with vertex colors based on the 'Group' attribute
plot(UKfaculty, vertex.color = vertex_colors)

UKfaculty network colored by Attribute group

Denotes the social phenomenon that people tend to be similar to their social contacts.

Birds of a feather - Billie Eilish has the same title as one of the most cited homophily papers by McPherson, Smith-Lovin and Cook (2001), where various studies of homophily were summarised. Homophily is observed in regard to race and ethnicity, gender and age, religion, education, occupation and social class, behaviour, and attitudes and beliefs

Previous research has found evidence of homophily regarding:

race and ethnicity (Moody 2001)
gender (Stehlé et al. 2013) and age
religion (eadleFriendshipDynamicsReligion2012?)
education, occupation, and social class
behaviour (e.g. political behaviour (Adamic and Glance 2005), sports, playing an instrument)
attitudes, abilities, beliefs

Definition: from what stage on, can we say a network is homophile or segregated?

Other, secondary types of homophily can reinforce segregation on an attribute. Example: Ethnic segregation can be reinforced by socioeconomic homophily, when ethnic origin (node color) and socioeconomic status (node shape) overlap (e.g., many ethnic minority members have a lower socioeconomic status and many ethnic majority members have a higher socioeconomic status)

For an explanation see (Wimmer and Lewis 2010, 588–94)

Measurement of segregation:

Let \(e_{AB}\) be the number of edges between two types of actors \(A\) and \(B\) (e.g. different faculties) and \(m\) be the total numbers of edges in the network. Then homophily exists if \(\frac{e_{AB}{m}\) is significantly lower than the probability of a random connection between two actors from type \(A\) and type \(B\).

Code

g <- data(UKfaculty)
UKfaculty

group1 <- V(g)[Group == 1]

We can also use the assortavity coefficient for measurement.

Assortavity coefficients are basically correlation coefficients that (in the case of categorical variables) can be measured by \(e_{i,j}\) being the ratio of edge from actors of type \(i\) to type \(j\)

\[ r = \frac{\sum_{i}e_{ii}-\sum_{i}a_ib_i}{1-\sum_{i}a_ib_i} \]

Code

# types are assumed to be integers starting with one
assortativity_nominal(UKfaculty, types = V(UKfaculty)$Group + 1)

or odds-ratios of within-group ties (Bojanowski and Corten 2014)

Code

orwg(UKfaculty, "Group")

telling us, that same-group tie odds are 12.866 times greater than tie odds between groups.

or Colemans Index (Coleman 1958) that compares the proportion of same-group neighbours to the proportion of that group in the network as a whole.

Code

coleman(UKfaculty, "Group")

which compares the proportion of same-group neighbors to the proportion of that group in the network as a whole. It is a number between -1 and 1. Value of 0 means these proportions are equal. Value of 1 means that all ties outgoing from a particular group are sent to the members of the same group. Value of -1 is the opposite – all ties are sent to members of other group(s).

Measurement of homophily

Measuring homophily preferences is difficult as we cannot directly observe preferences. In the social networks literature, the most common ways to estimate the strength of homophily preferences are Exponential Random Graph Models (ERGMs) and Stochastic Actor-Oriented Models (SAOMs). These estimate the existence of ties in dependence of actor attributes (such as similarity). While we will only be talking about these models in week 07, I still want to point out the caveat of ERGMs and SAOMs that they rely on a revealed preference assumption and infer preferences from the observed network structure.

Another approach to testing homophily preferences are permutation tests. A permutation is the result of a random reshuffle of the row and column names in an adjacency matrix. This creates a random network with the same structural properties (e.g., reciprocated ties) as the observed network. Who is connected to whom is, however different now. The level of segregation in the permuted network is now solely based on the network composition. If we permute the network a large number of times, we receive a distribution of segregation values expected due to the network composition. Then, we can test whether the observed segregation is statistically significant from the distribution of permutations. A significant difference tells us that preferences are (likely) the reason for the difference between the mean of the permutations and the observation. We, however, do not know which preferences are responsible!

Code

NPerm <- 100 # number of permutations

# create empty lists for the permuted matrices and the segregation measures
permlist <- as.list(1:NPerm)
seglist <- as.list(1:NPerm)

# set a random seed for replicability
set.seed(213)

for (perm in 1:NPerm) {# loop over all permutations
  
  # pick the original matrix
  mat <- friendmats_50[[1]]
  
  # re-sample (without replacement!) the names
  new_names <- sample(x = rownames(mat), 
                      size = length(rownames(mat)), 
                      replace = FALSE)
  
  # reassing the names
  rownames(mat) <- new_names
  colnames(mat) <- new_names
  
  # save the matrix
  permlist[[perm]] <- mat
  
  # create a graph
  g_perm <- graph_from_adjacency_matrix(adjmatrix = mat, 
                                        mode = "directed")
  # initialize smoking
  V(g_perm)$smoking <- NA
  
  for (ego in V(g_perm)$name) {
    
    # re-assign the smoking behavior based on the node name
    V(g_perm)$smoking[V(g_perm)$name == ego] <- dat_50$smoking[dat_50$idstud == ego]
  }
  
  # calculate the assortativity coefficient
  seglist[[perm]] <- assortativity_nominal(graph = g_perm, 
                                           types = V(g_perm)$smoking + 1)
  
}

# unlist the permuted assortativity coefficients
assorts <- unlist(seglist)

# test whether the mean of the distributed permutativities is likely to be the observed assortativity
t.test(x = assorts,
       mu = assortativity_nominal(g, types = V(g)$smoking + 1))

Social capital on the individual level

Structural approach

Vertex importance

Vertex importance

The structural approach on the individual level relates mainly to the works of Mark Granovetter (1973) and Ronald Burt (2009).

But before we dive into there theoretical work, the vertices of a graph can provide rich information about a network, its structure and its dynamics. In sociology and psychology contexts, this is particularly true, because more often than not vertices represent people. The fact that people play different roles and have different influences inside groups and communities has motivated centuries of sociological and psychological research, so it is unsurprising that the concept of vertex importance and influence is of great interest in the study of people or organizational networks.

It follows from much of the earlier work we have been doing in this book that the vertices of a graph can provide rich information about a network, its structure and its dynamics. In sociology and psychology contexts, this is particularly true, because more often than not vertices represent people. The fact that people play different roles and have different influences inside groups and communities has motivated centuries of sociological and psychological research, so it is unsurprising that the concept of vertex importance and influence is of great interest in the study of people or organizational networks.

But importance and influence are not precisely defined concepts, and to make them real within the context of graphs and networks we need to find some sort of mathematical definition for them. In many visual graph layouts, more important or influential vertices that have stronger roles in overall connectivity will usually be positioned toward the center of a group of other vertices. Intuitively therefore, we use the term ‘centrality’ to describe the importance or influence of a vertex in the connected structure of a graph.

Code

graph <- sample_gnm(n=14, m=17) 

plot(graph, vertex.color = "lavender")

IMAGE OF A GRAPH where each centrality has a different node with different measures

Degree centrality:

The degree centralty (or valence) of a vertex \(v\) is the number of edges connected to \(v\). Its thus a measure of immediate connections. Related to the concept of degree is the ego size. The \(n\)th order ego network of a given vertex \(v\) is the set including \(v\) itself and all vertices that are reachable from \(v\) by a path of length \(n\). The size of the \(n\) th order ego network is the number of vertices in it.

Out-degree: the number of edges going out from a vertex.

In-degree: the number of edges going into a vertex.

PLot of a small network with in degree as vertex label and size

Code

# degree centrality in r
degree(UKfaculty, mode = "all")
degree(UKfaculty, mode = "in")
degree(UKfaculty, mode = "out")

Closeness centrality:

The closeness centrality of a vertec \(v\) in a connected graph is the inverse of the sum of the distances from \(v\) to all other vertices.

Betweenness centrality

Eigenvektor centrality:

The Eigenvector centrality (or prestige) is a measure of how connected a vertex is to other influential vertices in the graph. It is impossible to define this without a little linear algebra.

Which node in the network has the highest

Degree centrality
Closeness centrality
Eigenvector centrality?

Aufgabe: Why your friends have more friends than you do (Feld 1991)

The strength of weak ties

Structural holes

Remember last sessions excurse on cutpoints and bridges?

What is a cutpoint?
What is a bridge?

Resource based aproach

This approach is closely linked to the name of Pierre Bourdieu 1982. He proposed that success depended on the level of capital individuals can mobilize. While Bourdieu remains a bit vage about what he means, Lin 2001 has some clearer definitions. But basically, the concept boils down to the access to resources through the social network. In the most simple terms we could operationalize this as the resources possessed by the (direct) social ties of ego.

Tie formation and Social Capital

Triadic closure

Triadic closure is a concept in social network theory that was first introduced by Georg Simmel. It describes the tendency of two nodes \(A\) and \(B\) to become connected if they share a common neighbor \(C\). It can be used to understand and predict the formation of social ties in network (of course, there are many other factors influencing tie formation).

The two most common measures of triadic closure for a graph are clustering coefficient and transitivity.

Clustering coefficient

A clustering coefficient is a measure of the degree to which vertices in a graph tend to cluster together. Evidence suggests that in most social networks, groups tend to create highly clustered subgroups.

Global clustering coefficient or transitivity:

In an undirected graph a connected triple is a path of length 2 (i.e. triad 201). A triangle is a cycle of exactly three nodes (i.e. triad 300). The global clustering coefficient is then defined as

\[ G_g = \frac{(\text{number of triangles} \times 3)}{(\text{number of connected triples})} \]

\(C_g = 0\): no clustering
\(C_g = 1\): all triples are triangles (i.e. all connected triples are triangles)

Local clustering coefficient: The local clustering coefficient gives an indication of the of the extend of clustering of a single vertex (How close its neighbours are to being a clique (complete graph).).

Let \(G = (V, E)\) be an undirected simple graph¹ with \(V\) vertices and \(E\) edges. \(n\) is the total number of vertices in the graph, \(m\) the total number of edges.

The neighbourhood \(N_i\) of a vertex \(v_i\) are its immediatle neighbours and defined as followed:

\[ N_i = \{v_j : e_{ij} \in E \wedge e_{ji} \in E\} \]

Whe define \(k_i\) as he number of vertives \(|N_i|\) in the neighbourhood of \(v_i\). The local clustering coefficient of a vertex \(v_i\) is the proportion of edges between the vertices within its neighbourhood divided by the number of edges that could possibly exist between them.

Thus we define the local clustering coefficient of a vertex \(v_i\) as:

\[ C_i = \frac{\{|e_{jk} : v_j, v_k \in N_i, e_{jk} \in E|\}}{k_i (k_1-1)} \]

where \(e_{ij}\) is the number of edges between the vertices in the neighbourhood of \(v_i\) and \(k_i\) is the number of vertices in the neighbourhood of \(v_i\).

Code

transitivity(UKfaculty, type = "global")
transitivity(UKfaculty, type = "local")

Cognitive balance theory

Graph from ww2

Aufgaben:

Laden Sie den Datensatz middleschool aus dem Paket UserNetR! Denken Sie an die Konventierung des Netzwerks zu einem igraph-Objekt (package intergraph)! Lassen Sie sich die Anzahl der Knoten und Kanten ausgeben! Ist dieses Netzwerk gerichtet? Berechnen Sie den Dyadenzensus, die Reziprozität, den Triadenzensus und die Transitivität (Clustering-Koeffizient)! Welche Unterschiede lassen sich bezüglich dieser Maßzahlen im Vergleich zum AIDS-Blog-Netzwerk festellen? Verwenden sie einen (oder zwei) data.frame, um die beiden Netzwerke zu vergleichen! Denken Sie daran, auf die Unterschiede in der Größe zu korrigieren (das AIDS-Blog-Netzwerk is viermal so groß, wie das Middleschool-Netzwerk)! Was könnten die Ursachen für die Unterschiede sein?

Strength of weak ties

Structural holes

References

Adamic, Lada, and Natalie Glance. 2005. “The Political Blogosphere and the 2004 U.S. Election: Divided They Blog.”

Bojanowski, Michał, and Rense Corten. 2014. “Measuring Segregation in Social Networks.” Social Networks 39 (October): 14–32. https://doi.org/10.1016/j.socnet.2014.04.001.

Burt, Ronald S. 2009. Structural Holes: The Social Structure of Competition. Harvard University Press. https://doi.org/10.4159/9780674029095.

Coleman, James S. 1958. “Relational Analysis: The Study of Social Organizations with Survey Methods.” Human Organization 17 (4): 28–36. https://www.jstor.org/stable/44124097.

Davis, James A., and Samuel Leinhardt. 1967. “The Structure of Positive Interpersonal Relations in Small Groups.”

Granovetter, Mark S. 1973. “The Strength of Weak Ties.” American Journal of Sociology 78 (6): 1360–80. https://www.jstor.org/stable/2776392.

Lin, Nan. 2001. Social Capital: A Theory of Social Structure and Action. 1st ed. Cambridge University Press. https://doi.org/10.1017/CBO9780511815447.

Moody, James. 2001. “Race, School Integration, and Friendship Segregation in America.” American Journal of Sociology 107 (3): 679–716. https://doi.org/10.1086/338954.

Putnam, Robert D. 2001. Bowling Alone: The Collapse and Revival of American Community. 1. Touchstone ed. A Touchstone Book. London: Simon & Schuster [u.a.].

Stehlé, Juliette, François Charbonnier, Tristan Picard, Ciro Cattuto, and Alain Barrat. 2013. “Gender Homophily from Spatial Behavior in a Primary School: A Sociometric Study.” Social Networks 35 (4): 604–13. https://doi.org/10.1016/j.socnet.2013.08.003.

Wimmer, Andreas, and Kevin Lewis. 2010. “Beyond and Below Racial Homophily: ERG Models of a Friendship Network Documented on Facebook.” American Journal of Sociology 116 (2): 583–642. https://doi.org/10.1086/653658.

Footnotes

A simple graph is a graph that does not contain multiple edges or loops.↩︎

Week 04

Relational Sociology and the concept of embeddedness

Reciprocity

Dyad census in a graph

Transitivity

Network segregation and homophily

Measurement of segregation:

Measurement of homophily

Strength of weak ties

Structural holes

References

Footnotes

Copyright

Relational Sociology and the concept of embeddedness

Reciprocity

Dyad census in a graph

Transitivity

Network segregation and homophily

Measurement of segregation:

Measurement of homophily

Social capital on the individual level

Structural approach

Vertex importance

The strength of weak ties

Structural holes

Resource based aproach

Tie formation and Social Capital

Triadic closure

Clustering coefficient

Cognitive balance theory

Strength of weak ties

Structural holes

Social capital on the group level

Structural approach

Resource based aproach

References

Footnotes

Copyright